Method and apparatus for generating profile of solutions trading off number of activities utilized and objective value for bilinear integer optimization models

ABSTRACT

A computer system for generating at least one of a solution and a profile of solutions for a problem includes an augmenting unit that repeatedly solves a model to generate a profile of solutions. The profile trades off a reduction of an objective of the problem against a number of distinct activities utilized in a solution.

The present application is a Divisional Application of U.S. patentapplication Ser. No. 12/098,172, filed on Apr. 4, 2008, which was aContinuation Application of U.S. patent application Ser. No. 11/068,861,filed on Mar. 2, 2005, the entire contents of which are incorporatedherein by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention generally relates to a method and apparatus forinteger linear programming, and more particularly to a method andapparatus for generating a profile of solutions that trades off a numberof activities utilized against an objective value for a bilinear integeroptimization model. The bilinear integer optimization model seeks todetermine the activities (encoded by integers) and the integerutilizations of these activities, so as to minimize the objective value.For clarity, the exemplary method (and system is developed and discussedin an exemplary case of a cutting stock problem. In this case, a profileof solutions is generated to trade off the number of patterns used tocut raw units of stock material while minimizing the amount of materialwasted.

2. Description of the Related Art

When attempting to solve a specific problem, it is desirable to providea profile of possible solutions as opposed to generating a singlepossible solution for the problem. This idea will be illustrated in thepresent application using the exemplary “cutting stock problem”. In thecutting stock problem, a user has a supply of stock rolls. The stockrolls may include any stock material that may be available in discreteunits including, but not limited to, paper, chemical fiber, film, steel,wire and cable, wood, etc. The material in stock rolls has a width ofW_(max). A certain number, d^(i) of stock rolls are demanded of materialwith width w^(i) (where i=1, 2, . . . , m; and w^(i)<W_(max)). The userdesires to cut up the stock rolls to satisfy the demand while minimizingthe amount of trim loss created.

There are several additional restrictions, which must be considered whenhandling the cutting stock problem. For instance, W_(min) is a lowerbound on the allowed nominal trim loss per stock roll. Additionally, thecutting machines used to cut the stock material have a limited number ofknives, which limits the type of patterns that may be utilized.Furthermore, widths of material up to a threshold a) are considered tobe “narrow”, and there is a limit on the number (C_(nar)) of “narrows”permitted to be cut from a stock roll of material.

It is disadvantageous to provide the user with a single solution to thecutting stock problem, or any other problem. It is desirable to providea profile of solutions that trades off a minimization of an objective ofthe problem against a number of activities utilized in the solution. Inthe case of the cutting stock problem, it is desirable to provide aprofile of solutions that trades off the amount of trim loss createdagainst the number of distinct cutting patterns used. In the cuttingstock problem, it is important to limit the number of distinct cuttingpatterns, because there is a changeover cost associated with setting upthe stock cutting machine to cut a different pattern.

Several column generation approaches are conventionally used forgenerating solutions to a proposed problem. One such approach is theGilmore and Gomory method (e.g., see Paul C Gilmore and Ralph E. Gomory,“A Linear Programming Approach to the Cutting-Stock Problem”, OperationsResearch, 9:849-859, 1961). In the Gilmore and Gomory method constraintsinvolving coverage of given demand using a known set of activities aretreated in a master problem. Constraints describing feasible activitiesare treated in a subproblem.

The Gilmore and Gomory approach seeks a linear-programming solution atthe master problem level. Thus, necessary information is communicated tothe subproblem by prices (i.e., dual variables). Since the user desiresan integer solution to the master problem, prices are not sufficient todescribe the optimal coverage of demand based on known activities to thesubproblem.

The above shortcoming of the conventional column-generating methods isovercome by “branching”. A common branching method is theBranch-and-Price method (e.g., see Gleb Belov and Guntram Scheithauer,“The Number of Setups (Different Patterns) in One-Dimensional StockCutting”, 2003, Preprint, Dresden University, Institute for NumericalMathematics). This method requires substantial computational resourcesas it emphasizes proving the optimality of a solution, as opposed tofinding a good approximate solution. Moreover, this method does not lenditself to efficiently providing a profile of solutions with the desiredtradeoff.

The Branch-and-Price method tends to provide good solutions, but becauseof the substantial computational requirements, can be a very slowprocess. On the other hand, the Gilmore and Gomory process is a fastprocess, but may not provide good overall solutions.

SUMMARY OF THE INVENTION

In view of the foregoing and other exemplary problems, drawbacks, anddisadvantages of the conventional methods and structures, an exemplaryfeature of the present invention is to provide a method and system inwhich a good compromise is achieved between the quality of the solutionobtained and the speed with which the solution is generated.

It is another exemplary feature to provide a method and system thatlimits the number of distinct activities utilized in the solution, whichwill reduce the overall cost of the solution.

It is another exemplary feature to provide a method and system whereinit is possible to generate new activities while the profile of thesolution is being generated.

To achieve the above and other features, in a first aspect of thepresent invention, a method (and system) of generating at least one of asolution and a profile of solutions for a problem, includes trading offa reduction of an objective of the problem against a number of distinctactivities utilized in a solution.

In a second aspect of the present invention, a method of generating atleast one of a solution and a profile of solutions for a cutting stockproblem, includes trading off a reduction of an amount of wasted stockmaterial against the number of distinct patterns utilized to satisfydemands.

In a third aspect of the present invention, a computer system forgenerating at least one of a solution and a profile of solutions for aproblem, includes partitioning means for generating a model having afirst set of variables and data and a second set of variables and dataand augmenting means for repeatedly solving the model to generate theprofile of solutions.

In a fourth aspect of the present invention, a computer system forgenerating at least one of a solution and a profile of solutions for aproblem, includes an augmenting unit that repeatedly solves a model togenerate a profile of solutions, wherein the profile trades off areduction of an objective of the problem against the number of distinctactivities utilized in a solution.

In a fifth aspect of the present invention, a signal-bearing mediumtangibly embodying a program of machine readable instructions executableby a digital processing apparatus to perform a method of generating atleast one of a solution and a profile of solutions for a problem,includes trading off a reduction of an objective of the problem againsta number of distinct activities utilized in a solution.

In a sixth aspect of the present invention, a method of deployingcomputing infrastructure, includes integrating computer-readable codeinto a computing system, wherein the computer readable code incombination with the computing system is capable of performing a methodof generating at least one of a solution and a profile of solutions fora problem, here the method of generating at least one of a solution anda profile of solutions for a problem, includes trading off a reductionof an objective of the problem against a number of activities utilizedin a solution.

The present invention builds a model having two types of activities. Thetwo types of activities include known activities and newly createdactivities. A working set of activities, or known activities, areprovided with associated variables, which determine the utilization ofthe known activities. There are further variables representing thepotentially new activities, and associated variables that determine theutilization of the potential new activities.

The present invention solves a sequence of models of activities andtransfers certain new activities into the set of known activities. Thisallows the set of known activities to be built to any desired number ofactivities. During the process, activities in the set of knownactivities may be removed if more desirable, new activities aregenerated.

Using binary expansion, linear expressions in binary (“0”/“1”) variablesare substituted for each of the variables (e.g., activity variables andassociated utilization variables) associated with the new activities. Astandard linearization is used to model the binary products throughlinear inequalities.

The present invention addresses certain bilinear integer optimizationmodels. The models addressed have a general form of:

Min c ^(t) x,+g ^(t) u subject to

Zx+Gu=d,

Hu≦f,

x,u≧0 and integer,

Z satisfying:

Az≦b, for all columns z of Z,

z≧0 integer, for all columns z of Z,

The number of columns of Z should be no greater than n_(max). Eachactivity produces certain outputs and the entries in a column z indicatethe outputs. For example z^(j) gives the number of units of the j-thoutput type produced by the activity associated with z.

The data of the model consists of the vectors c, g, d, f and b and thematrices A, G and H The vector d is a demand vector that we seek tocover. The matrix Z is a matrix of nonnegative integer variables. Thecolumns z of Z represent activities that we will carry out. The vectorof nonnegative integer variables x specifies the utilizations of theactivities represented by the columns of Z. Constraints determiningallowable columns z of Z are specified with the data A and b (i.e.,Az≦b, z≧0 integer). The cost vector c is associated with the activitiesrepresented by the columns of Z. The variables u are additionalvariables used to allow some modeling flexibility. The data g, f, G andH are used to build any constraints and linkages associated with theseadditional variables u.

The method (and system) of the present invention also includescombinatorially-motivated constraints to tighten the formulation. Thepresent method relies on the solution of a sequence of integer linearprograms. Standard methods for solving such problems rely on solvingfurther sequences of linear programming relaxations (where we ignore theinteger restrictions). The efficiency of the entire process depends onhow well the linear programming relaxation approximates the integerprogram (in terms of optimal objective value and nearness of thefeasible region of the linear programming relaxation to the smallestconvex set that contains the integer feasible points). Tightening theformulation refers to adjoining further linear inequalities that makethe linear programming relaxation a sharper approximation of the integerprogram. Additional linear constraints are provided to eliminatesymmetry between the new activities, and to preclude generating newactivities that are already included in the set of known activities.

Additionally, the method (and system) of the present invention includesthe facility to dynamically change the number of bits for representingutilizations of new activities. As new activities are generated, theiroptimal utilizations tend to decrease as we allow the total number ofactivities to increase. Moreover, as the number of activities increases,the difficult of the integer linear programs increase. Computationalefficiencies can be realized be allowing the number of bits to vary(usually decrease) dynamically.

The present invention addresses these and other shortcomings of theconventional methods by formulating a holistic model that simultaneouslyseeks a small number of new activities, their utilizations and theutilizations of the known set of activities. That is, the unified modelof the present invention generates new patterns in situ. Another benefitof the present invention is that it solves a sequence of models in alocal-search framework which is ideal for working with constraints thatare not modeled effectively in an integer linear programming (ILP)setting, but are relatively simple (e.g., a limit on the number ofactivities utilized), and (2) is well suited to efficiently generating aprofile of solutions trading off a secondary objective (e.g., the numberof distinct activities utilized) versus a primary objective.

With the above and other unique and unobvious exemplary aspects of thepresent invention, it is possible to balance the quality of a solutionand the speed in which the solution is generated.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other exemplary purposes, aspects and advantages willbe better understood from the following detailed description of anexemplary embodiment of the invention with reference to the drawings, inwhich:

FIG. 1 illustrates a sample Pareto curve graph comparing the amount oftrim loss to the number of distinct patterns utilized, for a profile ofsolutions satisfying given demands;

FIG. 2 illustrates an exemplary flow chart of a method 200 of generatinga profile of solutions trading off the number of activities utilized andthe objective value for bilinear integer optimization of the presentinvention;

FIG. 3 illustrates an exemplary block diagram of a system 300 forgenerating a profile of solutions trading off the number of activitiesutilized and the objective value for bilinear integer optimization ofthe present invention;

FIGS. 4A-4F show exemplary sample results obtained using the presentinvention;

FIG. 5 is an exemplary graph showing sample results obtained by thepresent invention in comparison with results obtained from aconventional method;

FIG. 6 is an exemplary graph illustrating a feature of the presentinvention for limiting the number of bits used in a binaryrepresentation for the usage of particular activity;

FIG. 7 illustrates an exemplary hardware/information handling system 700for incorporating the present invention therein; and

FIG. 8 illustrates a signal-bearing medium 800 (e.g., storage medium)for storing steps of a program of a method of the present invention.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS OF THE INVENTION

Referring now to the drawings, and more particularly to FIGS. 1-8, thereare shown exemplary embodiments of the method and structures accordingto the present invention.

To more clearly explain the method and structures for generating aprofile of solutions for a problem according to the present invention,the present invention will be described in the context of the exemplarycutting stock problem. The method (and system) of the present invention,however, may be used for generating a profile of solutions for otherdiscrete optimization problems (e.g., multicommodity flow andscheduling) by to optimizing the utilization of any activity, and aswould be evident to one of ordinary skill in the art taking the presentapplication as a whole, is not limited to optimizing the use of cuttingpatterns for cutting stock material.

Working with an integer bilinear programming formulation of aone-dimensional cutting-stock problem, the inventive method (and system)is an ILP-based local-search heuristic. The ILPs holistically integratethe master problem (e.g., constraints of the known patterns) and thesubproblem (e.g., constraints of the potential new patterns) of theusual price-driven pattern-generation paradigm, resulting in a unifiedmodel that generates new patterns in situ. The present method is wellsuited to practical restrictions such as when a limited number ofcutting patterns should be employed. In particular, the present methodand the results of computational experiments obtained from use of themethod are exemplarily described below in the context of a chemicalfiber cutting stock problem. The exemplary cutting method of the presentinvention may also be applied to cutting stocks of paper, film, steel,wire, cable, wood, etc. Again, however, the invention is not limited toproviding a solution only for the cutting stock problem.

The classical (one-dimensional) cutting stock problem (CSP) is tominimize the amount of trim loss created in covering the demands ofclient orders. The constant m represents the number of different widthsdemanded by the customers. The set M, which is used to index thedifferent widths demanded by the customers, is defined as M:={1,2, . . ., m}. Stock rolls of width W_(max) are available for satisfying customerorder demands d^(i) for rolls of width w^(i)(<W_(max)), iεM. Many modelsfor the cutting stock problem employ the idea of a cutting pattern. Apattern for cutting the stock material is defined by zεZ₊ ^(M)satisfying

$\begin{matrix}{{W_{\min} \leq {\sum\limits_{i \in M}\; {w^{i}z^{i}}} \leq W_{\max}},} & (1)\end{matrix}$

where W_(min) is a lower bound on the allowed nominal trim loss perstock roll (note that the total actual trim loss may be greater than thesum of the nominal trim losses, since in the method of the presentinvention, over-producing is permitted (i.e., that is, we are permittedto produce more than d^(i) for rolls of width w^(i), for iεM)). Thevariable z^(i) represents the number of rolls of width w^(i) that areobtained from a single stock roll when the pattern z is employed. Withthis concept, most models seek to determine which cutting patterns touse and how many times to repeat each pattern.

This is the framework of the conventional Gilmore and Gomory methoddiscussed above, which applied a simplex-method based column-generationscheme to solve the linear programming (LP) relaxation, followed byrounding (i.e., increasing any fractional pattern utilizations up to thenext integer). The present method allows for some variations of theconventional model to accommodate practical application.

In actual instances of the CSP, the cutting machines have a limitednumber (e.g., 10-20) of knives †. To account for this limitation, thefollowing restriction is added to the model:

$\begin{matrix}{{{\sum\limits_{i \in M}\; z^{i}} \leq {\dagger + \frac{\sum\limits_{i \in M}\; {w^{i}z^{i}}}{W_{\max}}}},} & (2)\end{matrix}$

or, equivalently,

$\begin{matrix}{{{\sum\limits_{i \in M}\; {( {W_{\max} - w^{i}} )z^{i}}} \leq {\dagger \; W_{\max}}},} & (3)\end{matrix}$

because the present method allows for †+1 pieces to be cut when there isno nominal trim loss for a pattern.

Also, widths up to some threshold ω are deemed to be narrow, and thereis often a limit on the number C_(nar) of narrows permitted to be cutfrom a stock roll (due to physical limitations of the cutting machines,it may not be possible to cut more narrow widths than some limit (e.g.,5)). To account for these limitations, the following restriction isadded to the model:

$\begin{matrix}{{{\sum\limits_{i \in M_{nar}}\; z^{i}} \leq C_{nar}},} & (4)\end{matrix}$where M _(nar) :={iεM:w _(i)≦ω}.

Additionally, the method of the present invention allows for thepossibility of not requiring the customer order demands to be exactlysatisfied. This is quite practical for real applications, for example inthe paper industry, where customers are willing to accept some variationin the quantities delivered from those ordered, and permitting this candecrease the trim loss. The method assumes that for some smallnonnegative integers q^(i) and p^(i) (e.g., q^(i) and p^(i) may be 5% ofthe demand d^(i)), the customer will accept delivery of betweend^(i)−q^(i) and d^(i)+p^(i) rolls of width w^(i), for iεM. The presentmethod allows for over-production, beyond d^(i)+p^(i) rolls of widthw^(i), but the over-production is treated as trim loss.

Finally, in the present invention, the number of patterns allowed islimited to some number n_(max). This type of limitation is practical foractual applications, but it is difficult to handle computationally.Heuristic approaches to this type of limitation include variations onthe Gilmore and Gomory approach discussed above, incremental heuristicsfor generating patterns, local-search methods for reducing the number ofpatterns as a post-processing stage, a linear-programming-based localsearch, and a heuristically-based local search. Also, computationallyintensive methods such as the Branch and Price method discussed abovehave also been applied. The present invention provides a hybrid methodthat is more powerful than existing heuristics (e.g., the Gilmore andGomory method), but with less effort than exact ILP approaches (e.g.,the Branch and Price method).

FIG. 1 illustrates a sample Pareto curve graph comparing the amount oftrim loss (e.g., scrap material) from a solution and the number ofpatterns used in the solution. When generating a solution to the cuttingstock problem, it is desired to minimize the amount of trim loss, whilesimultaneously minimizing the number of patterns used to cut the stockmaterial. Referring to the curve in FIG. 1, it is desired to locate apoint along the curve that provides the optimal trade-off between theamount of scrap material and the number of patterns.

FIG. 2 illustrates an exemplary flow chart of a method 200 of generatinga profile of solutions trading off the number of activities utilized andthe objective value for bilinear optimization according to the presentinvention.

First, initial activities, data and parameters are read in (step 201).Then a bilinear model is generated having two sets of activities (step202). The bilinear ILP model is based on a current set of activities Nand a number of potential new activities {circumflex over (N)}. In thecase of the cutting stock problem, N represents a set of known patterns,and {circumflex over (N)} represents a set of potential new patterns.The set of known patterns N is initially empty, and set of potential newpatterns {circumflex over (N)} is of a very limited size (e.g., no morethan 3 elements). However, N may include a relatively small number(e.g., between 5 and 10) of known patterns, as may be desired by theuser.

The bilinear model, of the present invention, is generated havingseveral limitations, as follows. The set N defined by N:={1,2, . . . ,n} is used to denote the set of patterns to be determined by thesolution procedure. A set of n patterns is defined by the columns of anm×n matrix Z. So, z^(ij) represents the number of rolls of width w^(i)that is cut from a stock roll when a pattern j is employed. The variablex^(j) is a nonnegative integer variable that represents the number oftimes the pattern j is employed (the utilization of the pattern j)toward satisfying the client order demands. The variable s^(i)represents the part of the deviation (from demand d^(i)) of theproduction of width w^(i) that is within the allowed tolerance. Thevariable t^(i) represents a part of the deviation that exceeds thetolerance and is treated as trim loss.

The bilinear model includes the following integer bilinear programmingformulation which seeks to minimize trim loss:

$\begin{matrix}{{{\min \; W_{\max}{\sum\limits_{j \in N}\; x^{j}}} - {\sum\limits_{i \in M}\; {w^{i}( {d^{i} + s^{i}} )}}}{{subject}\mspace{14mu} {to}}} & (5) \\{{W_{\min} \leq {\sum\limits_{i \in M}\; {w^{i}z^{ij}}} \leq W_{\max}},{{\forall{j \in N}};}} & (6) \\{{{\sum\limits_{i \in M}\; {( {W_{\max} - w^{i}} )z^{ij}}} \leq {\dagger \; W_{\max}}},{{\forall{j \in N}};}} & (7) \\{{{\sum\limits_{i \in M_{nar}}\; z^{ij}} \leq C_{nar}},{{\forall{j \in N}};}} & (8) \\{{z^{ij} \geq {0\mspace{14mu} {integer}}},{\forall{i \in M}},{{j \in N};}} & (9) \\{{{{\sum\limits_{j \in N}\; {z^{ij}x^{j}}} - s^{i} - t^{i}} = d^{i}},{{\forall{i \in M}};}} & (10) \\{{{- q^{i}} \leq s^{i} \leq p^{i}},{{\forall{i \in M}};}} & (11) \\{{t^{i} \geq 0},{{\forall{i \in M}};}} & (12) \\{{x^{j} \geq {0\mspace{14mu} {integer}}},{\forall{j \in {N.}}}} & (13)\end{matrix}$

For the most part, the constraints (5)-(13) include linear expressions(i.e., the sum of terms, each of which is the product of data and avariable). For example, constraint (6) multiplies the width of the stockroll w^(i) (fixed data) times the number of rolls z^(ij) (variable).Only constraint (10) includes a bilinear (variable times variable) term,z^(ij)x^(j). Constraints containing bilinear terms are extremelydifficult and time consuming to work with in an optimizationformulation, in comparison to linear terms, because in the bilinearcase, the feasible region of the relaxation (where we ignore integerrestrictions) is not generally a convex set.

There are a few strategies for dealing with the bilinearity ofconstraint (10). Allowing N to be so large that all possible patternscan be accommodated simultaneously, and applying decomposition in acertain manner, leads to the conventional linear-programming basedcolumn-generation approach of Gilmore and Gomory, which did not directlywork with the bilinear expressions.

The present invention, on the other hand, uses an alternative approach,which does not rely on decomposition, and thus does not involve anenormous number of variables. The method of the present inventionprovides a heuristic solution by partitioning N into two sets ofpatterns, N (known patterns) and {circumflex over (N)} (potential newpatterns).

For j N (the known patterns), the pattern variables z^(ij) (thepatterns) are treated as fixed data, and x^(j) the utilizations x^(j)are treated as true variables, for jε N. For jε{circumflex over (N)}(the potential new patterns), all of the variables x^(j) and z^(ij) arewritten in binary expansion and then the binary products are linearized.This results in an ILP model of the form (17)-(30).

Next, the ILP model is solved and new activities are determined whichare moved from {circumflex over (N)} to N (step 203).

After each move of new patterns to N, a local search is conducted (step204) of the potential new patterns and known patterns, testing whetherdropping a few patterns and re-generating patterns offers an improvement(i.e., decrease in the amount of trim loss) to the solution. This isaccomplished by solving further ILP models of the form (17)-(30).

After the local search is completed, we temporarily make {circumflexover (N)} empty and resolve the ILP model (step 205) to determineoptimal usages of all known activities. This may result in an improvedsolution if we had restricted the number of bits in the binary expansionof the utilizations x^(j) in the ILP models employed in steps 202 and204.

Then, the process determines if the maximum number of activities n_(max)has been reached (step 206). The process 200 is stopped when n_(max)patterns is reached (stop 207). If the maximum number of patterns hasnot been reached then the process is repeated (from step 202).

The linearization of the binary products, which occurs whenever ILPmodels of the form (17)-(30) with nonempty {circumflex over (N)} areinstantiated (steps 202 and 204) is described in detail below. Theparameter β^(i) represents an upper bound on the greatest number of bits(the number of times a particular pattern is used) used for representingz^(ij) in any solution to equations (6-9). For example,

β^(i):=┌log₂(1+min{└W _(max) /w ^(i)┘,└†/(1−w ^(i) /W _(max))┘})┐,∀iεM,M_(nar),

and

β^(i):=┌log₂(1+min{└W _(max) /w ^(i)┘,└†/(1−w ^(i) /W _(max))┘,C_(nar)})┐,∀iεM _(nar),

The parameter β represents an upper bound on the greatest number of bitsused for representing x^(j) in an optimal solution to the bilinearprogramming formulation (5-13). For example, β=(1+, where (x, z, s, t)is any feasible solution to (5-13). It is desirable to have the β^(i)and β small, since smaller values for these parameters generally lead toa decrease in computational effort. If the parameters are chosen toosmall, however, then the quality of the solution will be adverselyaffected (i.e., solutions with increased trim loss). The formulae givenfor β^(i) and β give the maximum values needed. Smaller values(potentially as small as 1) can be experimentally determined, to producereasonable run times without the quality of the solutions significantlydeteriorating.

The method defines L^(i):={0,1, . . . , β^(i)−1} for all iεM, andK:={0,1, . . . , β−1}: to index the bit variables in the binaryrepresentations of the integer variables of the model. The binaryvariables z_(l) ^(ij) are defined, for all iεM, jεN, lεL^(i), and thebinary variables x_(k) ^(j) are defined, for all jεN, kεK. These arejust the bits in the binary representations of the z^(ij) and the x^(j).For a given jεN, we refer to the z^(ij) as the integer-encoded pattern jand the z_(l) ^(ij) as the binary-encoded pattern j. So, letting

$\begin{matrix}{{z^{ij} = {\sum\limits_{l \in L^{i}}\; {2^{l}z_{l}^{ij}}}},{x^{j} = {\sum\limits_{k \in K}\; {2^{k}x_{k}^{j}}}},} & (14)\end{matrix}$

and

y _(kl) ^(ij) :=z _(l) ^(ij) x _(k) ^(j),  (15)

we have

$\begin{matrix}{y^{ij}:={{( {\sum\limits_{l \in L^{i}}\; {2^{l}z_{l}^{ij}}} )( {\sum\limits_{k \in K}\; {2^{k}x_{k}^{j}}} )} = {\sum\limits_{l \in L^{i}}\; {\sum\limits_{k \in K}\; {2^{l + k}{y_{kl}^{ij}.}}}}}} & (16)\end{matrix}$

Equations (14-16) are linearizations for modifying the bilinear modelfor making the transformation from the bilinear programming formulation(5-13) to the integer linear programming formulations (that weinstantiate in steps 202 and 204). The linearizations transform thebilinear terms into linear terms, at the expense of increasing thenumber of variables and introducing the inequalities (23). Thelinearizations are only used in the {circumflex over (N)} set ofpatterns (the terms involving patterns in N are linear since we considerthe associated patterns to be fixed). The linearizations aresubstitutions for removing the products z_(l) ^(ij), thereby to simplifythe computation.

The linearizations results in an inventive linear model having thefollowing ILP formulation:

$\begin{matrix}{{{\min \; {W_{\max}( {{\sum\limits_{j \in \overset{\_}{N}}\; x^{j}} + {\sum\limits_{j \in \hat{N}}\; {\sum\limits_{k \in K}\; {2^{k}x_{k}^{j}}}}} )}} - {\sum\limits_{i \in M}\; {w^{i}( {d^{i} + s^{i}} )}}}{{subject}\mspace{14mu} {to}}} & (17) \\{{W_{\min} \leq {\sum\limits_{i \in M}\; {w^{i}{\sum\limits_{l \in L^{i}}\; {2^{l}z_{l}^{ij}}}}} \leq W_{\max}},{{\forall{j \in \hat{N}}};}} & (18) \\{{{\sum\limits_{i \in M}\; {( {W_{\max} - w^{i}} ){\sum\limits_{l \in L^{i}}\; {2^{l}z_{l}^{ij}}}}} \leq {\dagger \; W_{\max}}},{{\forall{j \in \hat{N}}};}} & (19) \\{{{\sum\limits_{i \in M_{nar}}{\sum\limits_{l \in L^{i}}\; {2^{l}z_{l}^{ij}}}} \leq C_{nar}},{{\forall{j \in \hat{N}}};}} & (20) \\{{z_{l}^{ij} \geq {0\mspace{14mu} {integer}}},{\forall{i \in M}},{j \in \hat{N}},{{l \in L^{i}};}} & (21) \\{{{{\sum\limits_{j \in \overset{\_}{N}}\; {z^{ij}x^{j}}} + {\sum\limits_{j \in \hat{N}}\; {\sum\limits_{k \in K}\; {\sum\limits_{l \in L^{i}}\; {2^{k + l}y_{kl}^{ij}}}}} - s^{i} - t^{i}} = d^{i}},{{\forall{i \in M}};}} & (22) \\{{y_{kl}^{ij} \leq z_{l}^{ij}},{y_{kl}^{ij} \leq x_{k}^{j}},{{z_{l}^{ij} + x_{k}^{j}} \leq {1 + y_{kl}^{ij}}},{\forall{i \in M}},{j \in \hat{N}},{k \in K},{{l \in L^{i}};}} & (23) \\{{{- q^{i}} \leq s^{i} \leq p^{i}},{{\forall{i \in M}};}} & (24) \\{{t^{i} \geq 0},{{\forall{i \in M}};}} & (25) \\{{x^{j} \geq {0\mspace{14mu} {integer}}},{{\forall{j \in \overset{\_}{N}}};}} & (26)\end{matrix}$

The linear model (17)-(26) is instantiated and solved by the followingcomputational methodology of the present invention, which generates andsolves a sequence of instantiations that produce a profile of solutionsto the cutting stock problem (steps 202-207). For each iεM, jεN, thevariables and equations of (15) describe a Boolean quadric prototype.The ILP formulation (linear model) is strengthened (i.e., the feasibleregion of the linear programming relaxation is made smaller) using knownvalid inequalities. For example, some facets of the Boolean quadricpolytope are utilized as follows:

x _(k) ^(j) +z _(l) ^(ij) +y _(k′l′) ^(ij)≦1+y _(kl) ^(ij) +y _(kl′)^(ij) +y _(k′l) ^(ij),  (27)

y _(k′l) ^(ij) +y _(kl′) ^(ij) +y _(kl) ^(ij) ≦x _(k) ^(j) +z _(l) ^(ij)+y _(k′l′) ^(ij),  (28)

∀iεM,jε{circumflex over (N)},k,k′εK,l,l′εL ^(i).

Inequalities such as (27) and (28) are designed to restrict the solutionspace for the linear programming relaxation.

Pattern-Exclusion Constraints

$\begin{matrix}{{{{\sum\limits_{{i \in M},{{l \in {L^{i}:z_{l}^{i\; \overset{arrow}{\varphi}}}} = 0}}\; z_{l}^{ij}} + {\sum\limits_{{i \in M},{{l \in {L^{i}:z_{l}^{i\; \overset{arrow}{\varphi}}}} = 1}}( {1 - z_{l}^{ij}} )}} \geq 1},{\forall{j \in \hat{N}}},{\overset{\_}{\varphi} \in \overset{\_}{N}}} & (29)\end{matrix}$

are used in the model to insure that a binary-encoded pattern that isalready indexed in N is not re-generated as an integer-encoded pattern.The re-generation causes possible permutation symmetry that isdevastating to a branching procedure. The primary reason for encodingpatterns in binary is to accurately model multiplication. However, anadditional benefit is that patterns may be easily excluded from beingre-generated.

Additionally, implied integer-encoded patterns are ordered lexically, byimposing the following |{circumflex over (N)}|−1 lexicographicconstraints:

$\begin{matrix}{{{\sum\limits_{i \in M}\; {\sum\limits_{l \in L_{i}}\; {ɛ^{i}2^{l}z_{l}^{i\; 1}}}} \leq {\sum\limits_{i \in M}\; {\sum\limits_{l \in L_{i}}\; {ɛ^{i}2^{l}z^{i\; 2}}}} \leq \ldots \leq {\sum\limits_{i \in M}\; {\sum\limits_{l \in L_{i}}\; {ɛ^{i}2^{l}z^{i{\hat{N}}}}}}},} & (30)\end{matrix}$

for some ε>0 to overcome symmetry within {circumflex over (N)}. Choosingε sufficiently small (e.g., 0.001) insures a true lexical ordering, butthis causes numerical problems. A more moderate choice of ε (e.g., 0.25)is sufficient to break enough of the symmetry without causing numericaldifficulties. More sophisticated approaches for dealing with thissymmetry would be to try to apply further inequalities describing theall-different polytope.

A sequence of models of the form (17-30) is solved using the algorithmdescribed in FIG. 2. As discussed above, the set of patterns N ispartitioned into the set of known patterns N and the set of potentialnew patterns {circumflex over (N)} (step 201). In this initializationphase of the computational algorithm 200, N initially starts empty, ormay include a small number of patterns as desired by the user and{circumflex over (N)} allows for a small number v_(a) of potentialpatterns (e.g., ≦3).

Next, a model of the form (17)-(30) is instantiated (step 202). That is,variables are defined for the known patterns (indexed by N) and togenerate new patterns (indexed by {circumflex over (N)}). For jε N, thez^(ij) are treated as data, and the x_(j) are treated as true variables.For jε{circumflex over (N)}, the variables x^(j) and z^(ij) are writtenin binary expansion and then linearized.

Next, in the augmentation phase (step 203) of the algorithm 200, thesystem solves the model (17)-(30), and new patterns from the optimalsolution are transferred from {circumflex over (N)} to N.

Next, a local search is conducted to repeatedly test whether removingv_(s) (e.g., ≦3) patterns from N and adding v_(s) new patterns to N from{circumflex over (N)} through a new instantiation of the linear modelprovides an improvement in the solution (step 204). Once this localsearch terminates, an additional solve of the linear model is made todetermine the true optimal utilizations of all patterns in N (step 205).This may result in a further decrease in the trim loss, since we mayhave artificially restricted the number of bits used to represent theutilizations of the patterns that were just generated (in {circumflexover (N)}).

Next, the system tests whether the current number of patterns in N hasincreased to n_(max) (step 206). If it is, then algorithm 200 stops. Ifnot, then the algorithm continues from step 202.

FIG. 3 depicts an exemplary block diagram of a computer system 300 forgenerating a profile of solutions for a problem according to the presentinvention.

The computer system 300 includes at least an initializing unit 301 thatreads the data and sets the size of N and {circumflex over (N)}, aninstantiation unit 302 that instantiates linear models of the form(17)-(30), an augmenting unit 303 that solves instances of the linearmodel to determine new patterns and transfers them from N to {circumflexover (N)}, a local search unit 304 that solves a sequence of instancesof the linear model, successively determining whether dropping a patternfrom N and regenerating patterns through {circumflex over (N)} wouldoffer an improvement to the current solution, and an optimal utilizationunit 305 that solves an instance of the linear model directed atdetermining the optimal utilizations of all patterns N.

The method and system of the present invention are developed using theoptimization modeling/scripting software AMPL, and the followingexemplary results were obtained by conducting experiments on the methodand system of the present invention using the exemplary ILP solversCPLEX 9.0, Xpress-MP 14.27, and the open-source solver CBC (availablefrom www.coin-or.org). In place of AMPL the method can also utilize C,C++, Java, etc.

Again, the exemplary experiments were conducted in the context of achemical fiber cutting stock problem. The demands d^(i) having the samewidth w^(i)—which is common in these data sets, were aggregated forthese experiments. Note that in practice it may not be practical toperform such aggregation due to existing post-processing systems.

Each lower (resp., upper) demand tolerance q^(i) (resp., p^(i)) was setat five percent of demand, rounded down. The search parameters were setat v_(a)=v_(s)=1, which (after experimenting initially also withv_(a)=v_(s)=2) provided an adequately large search space at reasonablecomputational effort.

The formulation was strengthened by using linear inequalities for eachof the binary constraints (18-20) (e.g., minimal cover inequalities).Also, the pattern-exclusion constraints (29) may be aggregated andstrengthened to improve the computational efficiency.

In solving the ILPs, the method did not explicitly include all of theBoolean quadratic inequalities (28). Rather, the method solved the LP atthe root node using all of the inequalities (28). Then, the systemmaintained a few hundred of these that had the least slack at the LPoptimum and proceeded to solve the resulting ILP.

In the early iterations of the augmentation phase, when | N| is quitesmall, the ILP may not have a feasible solution. The method introducedan artificial variable into each of the demand constraints and penalizedthese variables in the objective function. Since each iteration of theaugmentation phase provides a feasible solution to the next iteration,after each such iteration the system permanently deleted any artificialvariables that had value 0. This step is very simple to carry out by oneof ordinary skill in the art, and it is beneficial in speeding upsubsequent iterations.

The method and system of the present invention was experimented with on39 chemical-fiber instances, and uniformly good solutions were obtained.Representative results are summarized in FIGS. 4A-4F.

In the example in FIG. 4A, the amount of trim loss is minimized when 11patterns are used to cut the stock material. However, as shown by thecurve, there is not a significant difference between the amount of trimloss when six patterns are used and when up to 11 patterns are used.Therefore, an optimal situation in this example would be to trade offthe small amount of trim loss and use 6 patterns to reduce the totalchangeover cost.

In the example in FIG. 4B, the amount of trim loss is minimized when 12patterns are used to cut the stock material. However, as shown by thecurve, there is not a significant difference between the amount of trimloss when 6 patterns are used and when up to 11 patterns are used.Therefore, an optimal solution in this example would be to trade off thesmall amount of trim loss and use 6 patterns to reduce the totalchangeover cost.

In the example in FIG. 4C, the amount of trim loss is minimized when 12patterns are used to cut the stock material. However, as shown by thecurve, there is not a significant difference between the amount of trimloss when 6 patterns are used and when up to 10 patterns are used.Therefore, an optimal situation in this example would be to trade offthe small amount of loss and use 6 patterns to reduce the totalchangeover cost. Also, when 8 or 9 patterns are used there is only 5%trim loss. The user may consider this an acceptable amount of trim loss,and therefore, use only 8 or 9 patterns to further reduce the changeovercosts.

In the example in FIG. 4D, the amount of trim loss is minimized when 10patterns are used to cut the stock material. However, as shown by thecurve, there is not a significant difference between the amount of trimloss when 7 patterns are used and when up to 10 patterns are used.Therefore, an optimal situation in this example would be to trade offthe small amount of trim loss and use 7 patterns to reduce the totalchangeover cost. Also, when 5 or 6 patterns are used there is less than5% trim loss. The user may consider this an acceptable amount of trimloss, and therefore, use only 5 or 6 patterns to further reduce thechangeover costs.

In the example in FIG. 4E, the amount of trim loss is minimized when 10patterns are used to cut the stock material. However, as shown by thecurve, there is not a significant difference between the amount of trimloss when 7 patterns are used and when up to 10 patterns are used.Therefore, an optimal situation in his example would be to trade off thesmall amount of loss and use 7 patterns to reduce the total changeovercost. Also, when 4-6 patterns are used there is less than 5% trim loss.The user may consider this an acceptable amount of trim loss, andtherefore, use only 4-6 patterns to further reduce the changeover costs.

In the example in FIG. 4F, the amount of trim loss is minimized when 9patterns are used to cut the stock material. However, as shown by thecurve, there is not a significant difference between the amount of trimloss when 6 patterns are used and when up to 9 patterns are used.Therefore, an optimal solution in this example would be to trade off thesmall amount of trim loss and use 6 patterns to reduce the totalchangeover cost.

The results of the present invention, as depicted in FIGS. 4A-4F providea user with a profile of solutions so that the user may determine thenumber of patterns to use by trading off the cost of using an additionalpattern against the amount of trim loss saved from using the additionalpattern.

FIG. 5 is an exemplary graph showing sample results 501 obtained by themethod of present invention in comparison with results 502 obtained froma conventional method. The result curve 501 of the present inventionprovides a profile of solutions, similar to those depicted in FIGS.4A-4F. The result curve 501 provides a user with an optimal solution forthe number of patterns to use, as well as allows the user to weigh thecosts and advantages of using additional patterns.

The results 502 obtained from using the conventional approaches,however, do not provide a profile of solutions. The conventionalapproaches provide the user with a single solution, in this case 20patterns, which does not allow the user to tradeoff the costs andadvantages of using additional patterns.

In this particular case, the conventional methods would have informedthe user that using 20 patterns would optimally minimize the amount oftrim loss, but as can be seen from the results 501 of the presentinvention using anywhere from 7-9 cutting patterns would not result in asignificant amount of trim loss, but would significantly reduce thechangeover cost associated with using 20 cutting patterns.

Therefore, with modest computational effort, the method and system ofthe present invention can obtain solutions with very little trim loss,using many fewer patterns than would be obtained using the conventionalmethods. Furthermore, the augmentation approach of the invention isespecially well suited for delivering a profile of solutions trading offthe number of patterns used against trim loss.

As it stands, the number of binary bit variables for these usages isβ|{circumflex over (N)}|, and there is a need to limit the number ofthese variables if one is to have some hope of quickly solving each ofthe ILPs along the way. The binary bit variables represent the number oftimes an activity is used. For example, if a bit variable x_(k) ^(j) isset equal to 1, that corresponds to the particular activity, or pattern,being used 2^(k) times.

The method of the present invention allows the user to artificiallylimit the number of binary bit variables. This is desirable for severalreasons. For example, the model can take larger values for |{circumflexover (N)}| if it takes β to be smaller than that required by the maximumpossible usage of a single pattern (in the computational experiments,β:=5). If a good pattern is found in this manner, then once it istransferred to {circumflex over (N)}, it will enjoy unrestricted usage.

FIG. 6 illustrates exemplary results obtained when β is in the rangefrom 2 up through 8. It can be readily seen that for this data set, verygood results are achieved with β is as low as 4 if the user can tolerate31 patterns—even though in the good solutions found, there are somepatterns that are used close to 200 (>>2⁴−1=15) times. For fewer numbersof patterns allowed, there is significant benefit to being more generouswith the bits. The best strategy seems to be one of dynamically varyingβ, starting fairly large and decreasing it as | N| increases.

For example, to achieve optimal results, while reducing the number ofbits used and the number of patterns used, a user should start out byemulating curve 8 until 31 patterns are used. At this point, the usercan switch to having 4 bits, because as can be seen from FIG. 6, theamount of trim loss is the same, once 31 patterns or more are used, for4 bits-8 bits. Then, once 36 patterns are used, the user could shift tohaving 2 bits. By providing the user with the plurality of solutionprofiles as depicted in FIG. 6, it allows the user to dynamically changethe number of bits being used to minimize the computation timeassociated with generating a good profile of solutions.

FIG. 7 shows a typical hardware configuration of an informationhandling/computer system in accordance with the invention thatpreferably has at least one processor or central processing unit (CPU)711. The CPUs 711 are interconnected via a system bus 712 to a randomaccess memory (RAM) 714, read-only memory (ROM) 716, input/outputadapter (I/O) 718 (for connecting peripheral devices such as disk units721 and tape drives 740 to the bus 712), user interface adapter 722 (forconnecting a keyboard 724, mouse-726, speaker 728, microphone 732,and/or other user interface devices to the bus 712), communicationadapter 734 (for connecting an information handling system to a dataprocessing network, the Internet, an Intranet, a personal area network(PAN), etc.), and a display adapter 736 for connecting the bus 712 to adisplay device 738 and/or printer 739 (e.g., a digital printer or thelike).

As shown in FIG. 7, in addition to the hardware and process environmentdescribed above, a different aspect of the invention includes acomputer-implemented method of performing the inventive method. As anexample, this method may be implemented in the particular hardwareenvironment discussed above.

Such a method may be implemented, for example, by operating a computer,as embodied by a digital data processing apparatus to execute a sequenceof machine-readable instructions. These instructions may reside invarious types of signal-bearing media.

Thus, this aspect of the present invention is directed to a programmedproduct, comprising signal-bearing media tangibly embodying a program ofmachine-readable instructions executable by a digital data processorincorporating the CPU 711 and hardware above, to perform the method ofthe present invention.

This signal-bearing media may include, for example, a RAM (not shown)contained with the CPU 711, as represented by the fast-access storage,for example. Alternatively, the instructions may be contained in anothersignal-bearing media, such as a magnetic data storage diskette or CDdisk 800 (FIG. 8), directly or indirectly accessible by the CPU 711.

Whether contained in the diskette 800, the computer/CPU 711, orelsewhere, the instructions may be stored on a variety ofmachine-readable data storage media, such as DASD storage (e.g., aconventional “hard drive” or a RAID array), magnetic tape, electronicread-only memory (e.g., ROM, EPROM, or EEPROM), an optical storagedevice (e.g., CD-ROM, WORM, DVD, digital optical tape, etc,), or othersuitable signal-bearing media including transmission media such asdigital and analog and communication links and wireless. In anillustrative embodiment of the invention, the machine-readableinstructions may comprise software object code, compiled from a languagesuch as “C”, etc.

Additionally, it should also be obvious to one of skill in the art thatthe instructions for the technique described herein can be downloadedthrough a network interface from a remote storage facility.

The present method (and system) addresses the shortcomings of theconventional pattern-generation approach by formulating a holistic modelthat simultaneously seeks a small number of new patterns, their usages,and the usages of the current set of patterns. That is, the unifiedmodel of the present invention generates new patterns in situ. Anotherbenefit of the unified model of the present invention is that it iseasily embedded in a local-search framework which is ideal for workingwith constraints that are not modeled effectively in an ILP setting.

While the invention has been described in terms of several exemplaryembodiments, those skilled in the art will recognize that the inventioncan be practiced with modification within the spirit and scope of theappended claims.

Further, it is noted that, Applicant's intent is to encompassequivalents of all claim elements, even if amended later duringprosecution.

1. A computer system for generating at least one of a solution and aprofile of solutions for a problem, said computer system comprising: anaugmenting unit that repeatedly solves a model to generate a profile ofsolutions, wherein said profile trades off a reduction of an objectiveof the problem against a number of distinct activities utilized in asolution.
 2. The computer system according to claim 1, furthercomprising: a partitioning unit that generates said model, said modelcomprising a first set of data and variables and a second set of dataand variables.
 3. The computer system according to claim 2, furthercomprising: an initializing unit that sets an initial size of said firstset of data and variables and an initial size of said second set of dataand variables.
 4. The computer system according to claim 2, furthercomprising: a testing unit that determines whether moving a point fromsaid second set of data and variables to said first set of data andvariables would improve the solution.
 5. A computer system forgenerating at least one of a solution and a profile of solutions for aproblem, said computer system comprising: means for generating a modelhaving a first set of variables and data and a second set of variablesand data; and means for repeatedly solving the model to generate aprofile of solutions.